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RedHat 9 (Linux i386) - man page for dstegr (redhat section l)

DSTEGR(l)					)					DSTEGR(l)

NAME
       DSTEGR  -  compute  selected eigenvalues and, optionally, eigenvectors of a real symmetric
       tridiagonal matrix T

SYNOPSIS
       SUBROUTINE DSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,  W,	Z,  LDZ,  ISUPPZ,
			  WORK, LWORK, IWORK, LIWORK, INFO )

	   CHARACTER	  JOBZ, RANGE

	   INTEGER	  IL, INFO, IU, LDZ, LIWORK, LWORK, M, N

	   DOUBLE	  PRECISION ABSTOL, VL, VU

	   INTEGER	  ISUPPZ( * ), IWORK( * )

	   DOUBLE	  PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE
       DSTEGR  computes  selected  eigenvalues	and, optionally, eigenvectors of a real symmetric
       tridiagonal matrix T. Eigenvalues and

	  (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
	      is a relatively robust representation,
	  (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
	      relative accuracy by the dqds algorithm,
	  (c) If there is a cluster of close eigenvalues, "choose" sigma_i
	      close to the cluster, and go to step (a),
	  (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
	      compute the corresponding eigenvector by forming a
	      rank-revealing twisted factorization.
       The desired accuracy of the output can be specified by the input parameter ABSTOL.

       For more details, see "A new O(n^2) algorithm  for  the	symmetric  tridiagonal	eigenval-
       ue/eigenvector  problem",  by Inderjit Dhillon, Computer Science Division Technical Report
       No. UCB/CSD-97-971, UC Berkeley, May 1997.

       Note 1 : Currently DSTEGR is only set up to find ALL the n eigenvalues and eigenvectors of
       T in O(n^2) time
       Note 2 : Currently the routine DSTEIN is called when an appropriate sigma_i cannot be cho-
       sen in step (c) above. DSTEIN invokes modified Gram-Schmidt when eigenvalues are close.
       Note 3 : DSTEGR works only on machines which follow ieee-754  floating-point  standard  in
       their  handling	of  infinities	and NaNs.  Normal execution of DSTEGR may create NaNs and
       infinities and hence may abort due to a floating point exception in environments which  do
       not conform to the ieee standard.

ARGUMENTS
       JOBZ    (input) CHARACTER*1
	       = 'N':  Compute eigenvalues only;
	       = 'V':  Compute eigenvalues and eigenvectors.

       RANGE   (input) CHARACTER*1
	       = 'A': all eigenvalues will be found.
	       =  'V':	all  eigenvalues in the half-open interval (VL,VU] will be found.  = 'I':
	       the IL-th through IU-th eigenvalues will be found.

       N       (input) INTEGER
	       The order of the matrix.  N >= 0.

       D       (input/output) DOUBLE PRECISION array, dimension (N)
	       On entry, the n diagonal elements of the tridiagonal matrix T. On exit, D is over-
	       written.

       E       (input/output) DOUBLE PRECISION array, dimension (N)
	       On entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in elements 1
	       to N-1 of E; E(N) need not be set.  On exit, E is overwritten.

       VL      (input) DOUBLE PRECISION
	       VU      (input) DOUBLE PRECISION If RANGE='V', the lower and upper bounds  of  the
	       interval  to  be searched for eigenvalues. VL < VU.  Not referenced if RANGE = 'A'
	       or 'I'.

       IL      (input) INTEGER
	       IU      (input) INTEGER If RANGE='I', the indices  (in  ascending  order)  of  the
	       smallest and largest eigenvalues to be returned.  1 <= IL <= IU <= N, if N > 0; IL
	       = 1 and IU = 0 if N = 0.  Not referenced if RANGE = 'A' or 'V'.

       ABSTOL  (input) DOUBLE PRECISION
	       The absolute error tolerance for the eigenvalues/eigenvectors. IF JOBZ = 'V',  the
	       eigenvalues and eigenvectors output have residual norms bounded by ABSTOL, and the
	       dot products between different eigenvectors are bounded by ABSTOL.  If  ABSTOL  is
	       less  than  N*EPS*|T|,  then N*EPS*|T| will be used in its place, where EPS is the
	       machine precision and |T| is the 1-norm of the tridiagonal matrix. The eigenvalues
	       are  computed  to  an accuracy of EPS*|T| irrespective of ABSTOL. If high relative
	       accuracy is important, set ABSTOL to DLAMCH( 'Safe minimum'  ).	 See  Barlow  and
	       Demmel  "Computing  Accurate Eigensystems of Scaled Diagonally Dominant Matrices",
	       LAPACK Working Note #7 for a discussion of which matrices define their eigenvalues
	       to high relative accuracy.

       M       (output) INTEGER
	       The  total  number of eigenvalues found.  0 <= M <= N.  If RANGE = 'A', M = N, and
	       if RANGE = 'I', M = IU-IL+1.

       W       (output) DOUBLE PRECISION array, dimension (N)
	       The first M elements contain the selected eigenvalues in ascending order.

       Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
	       If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the  orthonormal
	       eigenvectors  of  the matrix T corresponding to the selected eigenvalues, with the
	       i-th column of Z holding the eigenvector associated with W(i).	If  JOBZ  =  'N',
	       then  Z is not referenced.  Note: the user must ensure that at least max(1,M) col-
	       umns are supplied in the array Z; if RANGE = 'V', the exact  value  of  M  is  not
	       known in advance and an upper bound must be used.

       LDZ     (input) INTEGER
	       The  leading  dimension	of  the  array	Z.   LDZ  >= 1, and if JOBZ = 'V', LDZ >=
	       max(1,N).

       ISUPPZ  (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
	       The support of the eigenvectors in Z, i.e., the	indices  indicating  the  nonzero
	       elements  in  Z.  The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 )
	       through ISUPPZ( 2*i ).

       WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal (and minimal) LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK.  LWORK >= max(1,18*N)

	       If LWORK = -1, then a workspace query is assumed; the routine only calculates  the
	       optimal	size of the WORK array, returns this value as the first entry of the WORK
	       array, and no error message related to LWORK is issued by XERBLA.

       IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
	       On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

       LIWORK  (input) INTEGER
	       The dimension of the array IWORK.  LIWORK >= max(1,10*N)

	       If LIWORK = -1, then a workspace query is assumed; the routine only calculates the
	       optimal	size  of  the  IWORK  array, returns this value as the first entry of the
	       IWORK array, and no error message related to LIWORK is issued by XERBLA.

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value
	       > 0:  if INFO = 1, internal error in DLARRE,  if  INFO  =  2,  internal	error  in
	       DLARRV.

FURTHER DETAILS
       Based on contributions by
	  Inderjit Dhillon, IBM Almaden, USA
	  Osni Marques, LBNL/NERSC, USA

LAPACK computational version 3.0	   15 June 2000 				DSTEGR(l)


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